3
votes
1answer
40 views

tough lp inequalities

Let $1<p<\infty$. If possible, find a positive decreasing sequence $w_1>w_2>\cdots$ such that $\lim w_i=0$, and a (uniform) constant $K>0$, such that ...
4
votes
1answer
84 views

Proof that $(L^1)\neq(L^\infty)^\ast$

I have seen a "proof" that $L^1\neq(L^\infty)^\ast$ which goes as follows: show that there is an element of $(L^\infty)^\ast$ which is not in the image of the canonical map ...
1
vote
1answer
45 views

Banach valued sequence spaces $\ell^p(X)$

Let $X$ be a Banach space and $\ell^p(X)$ denote the space of sequences $x_i\in X$ for which the norm $\big(\sum_{i=1}^\infty\|x_i\|^p\big)^\frac1p$ is finite, when $X=\mathbb{R}$ we get the usual ...
3
votes
3answers
49 views

Not a basis for $ l^\infty$ then what is it?

We know that $ l^\infty$ has not a Schauder basis and its Hamel basis is uncountably infinite. Let $e_n=(e_{n1}, e_{n2},...)$ (for each $n\in \mathbb{N}$) s.t. $e_{nj}=0$ when $n\neq j$ and ...
4
votes
2answers
59 views

Reflexivity of $\ell^p$

I'm having bad difficulties in understanding how to prove that $\ell^p$ with $1<p<\infty$ are reflexive spaces. Every text I have consulted give that as a trivial result because "observing that ...
3
votes
1answer
89 views

Prove that $L^1(\mathbb{N})$ is a Banach space.

I'm trying to prove that $L^1(\mathbb{N}) := \left\{ (x_n)_{n=1}^{\infty} : \sum\limits_{n=1}^{\infty}\left|x_n\right| < \infty \right\} $, the space of all sequences over the field $\mathbb{C}$ ...
-1
votes
1answer
63 views

Example of open operator but not closed [closed]

Assume that $T:\ell_1\to\ell_2 $ is bounded,linear and one-to-one. Prove that $T(\ell_1)$ is not closed in $\ell_2$
0
votes
1answer
35 views

Isometric isomorphism between $\ell^{1}(\mathbb{R}^{3})$ and $\ell^{\infty}(\mathbb{R}^{3})$

Why is there no isometric isomorphism between $\ell^{1}(\mathbb{R}^{3})$ and $\ell^{\infty}(\mathbb{R}^{3})$? I know that there is such an isomorphism if $\mathbb{R}^{3}$ is replaced with ...
-2
votes
1answer
113 views

Continuity of $|.|$ in $W^{1,p}_0$

please i dont understand this proof We suppose that $u\rightarrow u$ on $W^{1,p}_0$ i dont understand why we must use the weak compactness and the uniform convexity of $W^{1,p}_0$ ? Thank you
5
votes
1answer
137 views

If $P$ has marginals $P_1, P_2$, is $L^1(P_1) + L^1(P_2)$ closed in $L^1(P)$?

Suppose that $\mathbb{X}=\mathbb{X}_1\times \mathbb{X}_2$ and suppose that $ P$ is a probability measure on $\mathbb{X}$ with marginals $ P_i$ on $\mathbb{X}_i, i=1,2$, i.e., $$\int f_i(x_i)\, ...
2
votes
1answer
48 views

Projection onto finite-dimensional subspace of $L^p$

Let $a_i$ be a basis of $L^p(\Omega)$ and consider $A_n = \text{span}\{a_1, ..., a_n\}$. Take an element $f \in L^p$. We want to define a projection onto the finite-dimensional subspace $A_n$. How do ...
2
votes
1answer
32 views

Exercise on L^p spaces

Let $f$ be a function of $L^p([0,2]) \>\> \forall p \in [1, \infty )$ and suppose $||f||_p \leq 1$. Show that $f$ belongs to $L^{\infty}([0,2])$ and $||f||_{\infty} \leq 1$.
0
votes
2answers
14 views

Sequence of continuous functions with unitary norm

I want to construct a sequence $f_n$ of continuous functions in $[0,1]$ such that $||f_n||=1$ (so a bounded sequence) and $||f_n-f_m||=1$ (it doesn't have any convergent subsequences). The norm is ...
0
votes
1answer
54 views

Am I wrong ? (2)

Let $X=C[0,1]$ be the space of real continous functions on $[0,1]$. $X$ is a Banach space with the two norms $$|f|_\infty=\sup_{s\in[0,1]}|f(s)|$$ and ...
1
vote
1answer
62 views

Is Lp space complete with this norm?

Let $E$ be a measurable set of finite measure and $1\leq a<b<\infty$. Consider the $L^b(E)$ space normed by $L^a$ norm. Is this space a Banach space? I think this is wrong, so I tried to find a ...
1
vote
1answer
46 views

Nontrivial functionals on $l^\infty$ vanishing on $c_0$

I understood that the dual of $c_0$ is a proper subspace of the dual of $l^\infty$, by Hahn-Banach theorem. But how can I find functionals in $(l^\infty)^*$ vanishing on $c_0$?
2
votes
1answer
63 views

the principle of uniform boundedness and $l^p$ space

If $1<p<\infty$ and $\{x_n\}\subset l^p$, then $\sum_{j=1}^\infty x_n(j)y(j)\to 0$ for every $y\in l^q$, $\frac{1}{p}+\frac{1}{q}=1$, iff $\sup_n||x_n||_P<\infty$ and $x_n(j)\to 0$ for every ...
1
vote
1answer
57 views

1<p<q, norm inequalities

Prove or disprove the following. (Recall that $c_{00}$ is the space of sequences with only finitely many nonzero entries.) Conjecture. Let $1<p<q<\infty$. Then there exists a function ...
2
votes
1answer
78 views

Two questions on Banach-valued spaces of integrable functions

Let $V$ be a Banach space with dual $V'$ and suppose that $V$ is included into the Hilbert space $H$ so that the inclusion is continuous and dense. Then after identification $H = H'%$ we have $V ...
1
vote
0answers
48 views

the dual space of $L^p$ [duplicate]

I am reading some preliminary material to develop a good background in order to study PDE and I came across the following fact The dual space of $L^p$ is $L^q$ where $q$ is the Holder's Conjugate of ...
1
vote
0answers
36 views

Examples of semigroups of contractive Fourier multipliers but not positive?

Can you show me a concrete an example of semigroup $(T_t)_{t\geq 0}$ of Fourier multipliers such that each operator $T_t$ induces a contractive Fourier multiplier $T_t\colon L^p(\mathbb{T}) \to ...
2
votes
0answers
41 views

uniform equivalence to unit vector basis of $\ell_p$

Let $(e_n)$ be the unit vector basis of $\ell_p$, $1\leq p<\infty$. It is well-known that if $(x_n)\subset\ell_p$ is seminormalized and weakly null then it contains a subsequence equivalent to ...
2
votes
1answer
47 views

$(\ell_p,\|.\|_{\infty})$ Banach or separable

Is $(\ell_p,\|.\|_{\infty})$ for $1\leq p<\infty$ a Banach or separable space? is there any fast way of proving it without checking separability or completeness with the usual way?
2
votes
1answer
37 views

$f_{n}$ converges in $L^{1}$ and $\|g_{n}\|_{L^{1}(\mathbb R)}\leq \|f_{n}\|_{L^{1}(\mathbb R)}\implies {g_{n}}$ converges in $L^{1}(\mathbb R)$?

Let $f_{n}, g_{n}\in L^{1}(\mathbb R)$ (Lebesgue space). Suppose there exist $f\in L^{1}(\mathbb R)$ such that $\|f_{n}-f\|_{L^{1}(\mathbb R)}\to 0$ as $n\to \infty;$ that is, the sequence $\{f_{n}\}$ ...
1
vote
0answers
32 views

A useful criterion in dispersive PDE.

I would like to prove the following theorem: Consider two Banach spaces $X\hookrightarrow Y$ and $1<p,q\leq\infty$. Let $(f_n)_{n\geq 0}$ be a bounded sequence in $L^q(I,Y)$ and let $f:I\mapsto Y$ ...
2
votes
2answers
57 views

is the following set complete in $l_p$?

I'm not sure what is the correct term in English, but by 'complete set' I mean that the span (of finite combinations) of the set is dense in the space. I have to show for which p ($ 1\leq p ...
19
votes
2answers
435 views

In $\ell^p$, if an operator commutes with left shift, it is continuous?

Our professor put this one in our exam, taking it out along the way though because it seemed too tricky. Still we wasted nearly an hour on it and can't stop thinking about a solution. What we have: ...
2
votes
1answer
41 views

property of a given subset $ A $ of $\mathit{l}^2 $

I was given this exercise Let $\lbrace e_n \rbrace $ be the canonical basis of $\mathit {l}^2 $ (the space of square sommable series with the usual norm) and set $$ A:= \lbrace \sum_{n \ in ...
2
votes
2answers
137 views

Differences between $L^p$ and $\ell^p$ spaces

Could someone explain some differences between the $L^p$ and $\ell^p$ spaces? Thanks a lot.
7
votes
4answers
369 views

$\ell^p$ is not isometric to $\ell^q$

The problem is this: if $1\le p<q<\infty$ then $\ell^p$ and $\ell^q$ are not isometric (as Banach spaces). This is an exercise but I'd like to see an elegant proof.
3
votes
1answer
53 views

Convergence in $L^1$ of a sequence of functions

I have to see if the following sequence of functions is convergent in the space $L^1[(0,\infty)]$ $$f_n(x)= n\frac{\exp\left(-\frac{n}{2x^2}\right)}{x^3}$$ By definition, $f_n(x)$ is convergent in ...
0
votes
1answer
150 views

characterization of weakly convergent to zero sequences on $l^p$ for $1\le p < \infty$

Let $1\le p< \infty$. Show that a sequence $t_k = ({t_{kj}})_{j=1}^{\infty}\in l^p$ converges weakly to 0 iff $||t_k||_p$ is bounded and $\lim_k t_{kj}=0$. I proved that if $t_k$ converges weakly ...
2
votes
0answers
214 views

Proving the $l_p$ space is complete.

I'm trying to prove $l_p$ spaces are complete. We have an $l_p$ space $W$. Let us take a cauchy sequence. There exists $N_0\in\Bbb{N}$ such that for $m,n>N_0$, $d(x^m,x^n)<\epsilon$. This ...
1
vote
1answer
240 views

Proof of existence of Schauder basis for $L^p(\Omega)$?

There are a statements around, see [Brezis 2011, p. 146], like All classical (separable) Banach spaces used in analysis have a Schauder basis . I was wondering where to find a proof confirming ...
1
vote
1answer
532 views

Doubt in the proof that $l^{p}$ is complete

I was looking at the proof that $l^{p}$ is complete with respect to the standard metric. Suppose $x^{(n)}$ is a Cauchy sequence in $l^{p}$. Then Given $\epsilon > 0$, $\exists\,\, n_{0} \in ...
0
votes
2answers
65 views

function in $L^1\setminus L^2$

I'm looking for an example of a function which belongs to the Banach space $L^1$ (i.e $\int|f|< \infty$) but is not in $L^2$ (so $\int|f|^2$ is unbounded). Does anyone know such a function?
3
votes
0answers
77 views

Mapping $G$ into its group algebra as left multiplication. Continuous?

I am reading an appendix on Group algebras which contains the following Proposition which I am trying to prove: Proposition: Let $G$ be a locally compact group, with $\zeta\in L^{p}(G)$ fixed. ...
5
votes
1answer
159 views

Is $(l^1 ,\|.\|)$ a Banach space?

Suppose $x=\{x_n\}\in l^1$ and $\|x\|=\sup|\sum_{k=1}^{n}x_k|$, let $\|x\|_1=\sum_{n=1}^{\infty}|x_n|$ is a norm for $l^1$ . Is $(l^1 ,\|.\|)$ a Banach space?
12
votes
1answer
132 views

Ideals of the algebra of all bounded linear operators on $\ell_p \oplus \ell_q$

Let $\mathcal{L}(X)$ be the algebra of all bounded linear operators from $X$ to $X$ for Banach space $X$. I need to show that $\mathcal{L}(\ell_p \oplus \ell_q)$ for $p \neq q$ contains at least two ...
4
votes
2answers
271 views

Completeness proof of $\ell^p$

Say $\{x_n\}$ is Cauchy in $\ell^p$ and $x$ is its pointwise limit. To argue that $x \in \ell^p$ would the following be correct: Let $\varepsilon > 0$ and let $N$ be s.t. $n,m > N$ ...
0
votes
2answers
123 views

How to compute the norm of this particular bounded linear functional?

On the Hilbert space $l^2$, let $f$ be the functional defined by $$f(x):= \sum_{j=1}^\infty \alpha_j \xi_j$$ for each $x:=(\xi_j)_{j=1}^\infty$ in $l^2$, where $a:= (\alpha_j)_{j=1}^\infty$ is a fixed ...
1
vote
0answers
501 views

Rademacher function and weak convergence

The function $r_{n}:[0,1]\rightarrow \{-1,1\}$ be defined by $r_{n}(t)=\operatorname{sgn}(\sin(2^{n}\pi t))$ is known as the $n$-th Rademacher function.a) Show that $r_{n}\xrightarrow{w}0$ in ...
1
vote
1answer
46 views

there is $M<\infty$ such that $\sum_{n} |\hat{f}(n)|\le M\int_{0}^{2\pi}|f(t)|dt$ for each $f\in X$

for $f\in L^1[0,2\pi]$ define $$\hat{f}(n)=\int_{0}^{2\pi} f(t)e^{-int} dt$$ for $n\in\mathbb{Z}$, $X$ is a closed linear subspace of $L^1[0,2\pi]$ such that $\sum_{n} |\hat{f}(n)|<\infty$ for each ...
2
votes
1answer
130 views

Closed subspace of $L^1[0,1]$

The statement I need to prove is following. Let $S$ be a closed subspace of Lebesgue space $L^1[0,1].$ Assume that for every $f\in S$ there exists a number $p(f)>1$ such that $f\in L^{p(f)}[0,1].$ ...
2
votes
1answer
165 views

weak vs. norm compactness in $\ell_1$

So I'm trying to show that weakly compact sets in $\ell_1$ are norm-compact. I've already proven that weak sequential convergence implies norm convergence. I think the idea I want to go with is to ...
1
vote
1answer
177 views

prove a subset of squence space $l^p$ closed in strong topology

Let $l^p$ be the space of $p$-summable sequences. von Neumann constructed a subset of $l^p$ space $$S=\{X_{mn}: m,n≥1\}$$ where $X_{mn}\in l^p$ are defined by $X_{mn}(m)=1, X_{mn}(n)=m$ and ...
1
vote
1answer
115 views

Isometric embeddings of $\ell_q^m$ into $\ell_p$ and $L_p$ for $p,q\in[1,+\infty]$

I'm looking for articles describing (or proving nonexistence) of isometric embeddings of $m$-dimensional space $\ell_q^m$ into $L_p$ and $\ell_p$ for $q,p\in[1,+\infty]$. Since $\ell_q^m$ is finite ...
7
votes
1answer
379 views

Isometry between $L_\infty$ and $\ell_\infty$

It is known that there exist some isomorphism between $L_\infty$ and $\ell_\infty$, which is not explicit at all. Could someone tell me whether there exist an isometric isomorphism between ...
4
votes
3answers
556 views

Cesàro operator is bounded for $1<p<\infty$

The Cesàro operator $T\colon \ell_{p}\to\ell_{p}$ is defined by $(Tx)_{k}=\frac{1}{k}\sum_{j=1}^{k}x_{j},\: k\in\mathbb{N}$, where $x=(x_{k})_{k=1}^{\infty}$ Show that $T$ is bounded if ...
6
votes
1answer
571 views

How do you prove that $\ell_p$ is not isomorphic to $\ell_q$?

I guess that for all $1\le p,q<\infty $, such that $p\ne q$ , the spaces $\ell_p$ and $\ell_q$ are not isomorphic, but how do you prove this?